Octave 3.8, jcobi/3

Average Error: 3.4 → 0.3

Time: 32.7s

Precision: binary64

Cost: 1732

\[\alpha > -1 \land \beta > -1\]

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(\alpha - -1\right) \cdot \frac{\beta + 1}{\beta + \left(3 + \alpha\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.5e+116)
     (/ (* (- alpha -1.0) (/ (+ beta 1.0) (+ beta (+ 3.0 alpha)))) (* t_0 t_0))
     (/ (+ (/ 1.0 beta) (/ alpha beta)) (+ beta (+ alpha 3.0))))))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.5e+116) {
		tmp = ((alpha - -1.0) * ((beta + 1.0) / (beta + (3.0 + alpha)))) / (t_0 * t_0);
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / ((alpha + beta) + (2.0d0 * 1.0d0))) / ((alpha + beta) + (2.0d0 * 1.0d0))) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.5d+116) then
        tmp = ((alpha - (-1.0d0)) * ((beta + 1.0d0) / (beta + (3.0d0 + alpha)))) / (t_0 * t_0)
    else
        tmp = ((1.0d0 / beta) + (alpha / beta)) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.5e+116) {
		tmp = ((alpha - -1.0) * ((beta + 1.0) / (beta + (3.0 + alpha)))) / (t_0 * t_0);
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)

↓

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.5e+116:
		tmp = ((alpha - -1.0) * ((beta + 1.0) / (beta + (3.0 + alpha)))) / (t_0 * t_0)
	else:
		tmp = ((1.0 / beta) + (alpha / beta)) / (beta + (alpha + 3.0))
	return tmp

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.5e+116)
		tmp = Float64(Float64(Float64(alpha - -1.0) * Float64(Float64(beta + 1.0) / Float64(beta + Float64(3.0 + alpha)))) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
end

↓

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.5e+116)
		tmp = ((alpha - -1.0) * ((beta + 1.0) / (beta + (3.0 + alpha)))) / (t_0 * t_0);
	else
		tmp = ((1.0 / beta) + (alpha / beta)) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5e+116], N[(N[(N[(alpha - -1.0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.50000000000000013e116

   1. Initial program 0.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \left(\alpha \cdot \beta + \left(\beta + 1\right)\right)}{\beta + \left(\alpha + 3\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
      Proof

      [Start] 0.1	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-47 [=>] 0.2	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-44 [=>] 0.1	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 0.1

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 3\right)} + 0}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   4. Simplified 0.1

      \[\leadsto \frac{\color{blue}{\left(\alpha - -1\right) \cdot \frac{\beta + 1}{\beta + \left(3 + \alpha\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      Proof

      [Start] 0.1	\[ \frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 3\right)} + 0}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-4 [=>] 0.1	\[ \frac{\color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\alpha + \left(\beta + 3\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-49 [=>] 0.1	\[ \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 3\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-17 [=>] 0.1	\[ \frac{\color{blue}{\left(\alpha - -1\right)} \cdot \frac{\beta + 1}{\alpha + \left(\beta + 3\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      rational.json-simplify-41 [=>] 0.1	\[ \frac{\left(\alpha - -1\right) \cdot \frac{\beta + 1}{\color{blue}{\beta + \left(3 + \alpha\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   if 2.50000000000000013e116 < beta

   1. Initial program 8.8

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   2. Simplified 11.3

      \[\leadsto \color{blue}{\frac{\frac{\beta + \left(\alpha + \left(\alpha \cdot \beta + 1\right)\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}}{\beta + \left(\alpha + 3\right)}} \]
      Proof

      [Start] 8.8	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-47 [=>] 11.3	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      rational.json-simplify-44 [=>] 11.2	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      rational.json-simplify-1 [=>] 11.2	\[ \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-17 [=>] 11.2	\[ \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) - -1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-50 [=>] 11.2	\[ \frac{\color{blue}{\frac{-\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}{-1 - \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-50 [=>] 11.2	\[ \frac{\color{blue}{\frac{-\left(-\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) - -1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-17 [<=] 11.2	\[ \frac{\frac{-\left(-\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right)}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-1 [<=] 11.2	\[ \frac{\frac{-\left(-\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      rational.json-simplify-44 [=>] 11.3	\[ \color{blue}{\frac{\frac{-\left(-\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

   3. Taylor expanded in beta around inf 0.6

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\beta + \left(\alpha + 3\right)} \]

   4. Taylor expanded in alpha around 0 0.6

      \[\leadsto \frac{\color{blue}{\frac{1}{\beta} + \frac{\alpha}{\beta}}}{\beta + \left(\alpha + 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(\alpha - -1\right) \cdot \frac{\beta + 1}{\beta + \left(3 + \alpha\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1604

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 165000000:\\ \;\;\;\;\frac{\frac{\beta + \left(\alpha + 1\right)}{t_0 \cdot t_0}}{\beta + \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(4 + 2 \cdot \alpha\right)}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1600

\[\begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ \frac{\frac{1 + \alpha}{t_0}}{\frac{t_0}{\frac{\beta + 1}{\left(\beta + 3\right) + \alpha}}} \end{array} \]

Alternative 3
Error	2.0
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.2222222222222222 \cdot \beta + 0.3333333333333333}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 4
Error	1.9
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) + 1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(6 + 5 \cdot \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	1220

\[\begin{array}{l} t_0 := \frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}\\ \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{t_0}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\beta + \left(4 + 2 \cdot \alpha\right)}\\ \end{array} \]

Alternative 7
Error	2.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 8
Error	2.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 9
Error	4.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 1.85 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 10
Error	2.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 11
Error	29.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 12
Error	31.9
Cost	448

\[\frac{1}{\beta \cdot \left(\beta + 3\right)} \]

Alternative 13
Error	31.6
Cost	448

\[\frac{\frac{1}{\beta}}{\beta + 3} \]

Alternative 14
Error	60.2
Cost	192

\[\frac{0.3333333333333333}{\beta} \]

Reproduce

herbie shell --seed 2023075 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))